On generic properties of closed geodesics

D. V. Anosov

Izvestiya. Mathematics , Tome 21 (1983) no. 1, pp. 1-29

Résumé

A complete proof is given of the theorem asserting that bumpy metrics are generic. This result was announced by Abraham (Global Analysis (Proc. Sympos. Pure Math., vol 14), Amer. Math. Soc., Providence, R. I., 1970, pp. 1–3.). Related results are used to rigourously carry out Poincaré's outline of a “bifurcation-theoretic” proof of the existence of closed geodesics without self-intersections for any Riemannian metric of positive curvature on the two-dimensional sphere $S^2$. To do this, it is essential that the lengths of all non-self-intersecting closed geodesics for the metrics on $S^2$, considered in the course of the proof, be uniformly bounded from above. Examples are given of $C^\infty$ metrics on $S^2$ (where the sign of the curvature alternates) for which there exist arbitrarily long closed geodesics without self-intersections. Bibliography: 27 titles.